Recall that \(W_+,W_+^*\in \mathbb {B}(L^p({\mathbb {R}}))\)  for all \(1<p<\infty \)  by Theorem REF  (1). Since \(f\in H^s({\mathbb {R}})\subset L^\infty ({\mathbb {R}})\)  for \(s>1/2\) , we thus obtain by [Formula of multiplier]Formula of multiplier that
\({\Vert f(H)\Vert }_{L^p\rightarrow L^p}\lesssim {\Vert f\Vert }_{L^\infty ({\mathbb {R}})}\sum _{j=1}^N{\Vert P_{\lambda _j}\Vert }_{L^p\rightarrow L^p}+{\Vert f(\Delta ^2)\Vert }_{L^p\rightarrow L^p}.\) 
In order to deal with the term \(f(\Delta ^2)\) , we let \(\widetilde{\eta }(\xi )=\eta (\xi ^4)\)  and \(m(\xi )=f(\xi ^4)\)  so that \(\eta (\xi ^4)f(\xi ^4)=\widetilde{\eta }(\xi )m(\xi )\)  and thus \(f(\Delta ^2)=m(D)\) . By Hörmander's condition [Hormanderconditions]Hormanderconditions, we have
\(\sup _{\delta >0}\Vert \widetilde{\eta }(\cdot )m(\delta \cdot )\Vert _{H^{s}(\mathbb {R})}\le C_s M<\infty \) 
with some \(C_s>0\)  independent of \(m,M\) , which implies by the classical Hörmander Fourier multiplier theorem (see [1]} or Grafakos [2]}) that
\(\Vert f(\Delta ^2)\Vert _{L^p\rightarrow L^p}=\Vert m(D)\Vert _{L^p\rightarrow L^p}\lesssim M+{\Vert f\Vert }_{L^\infty },\quad 1<p<\infty .\) 
It remains to show \(P_{\lambda _j}\in \mathbb {B}(L^p({\mathbb {R}}))\)  for each \(1\le j\le N\) . In fact, we just need to show the eigenfunction \(e_j(x)\)  belongs to \(L^p\)  for all \(1\le p\le \infty \)  since
\(\Vert P_{\lambda _j}\phi \Vert _{L^p}\le |\langle \phi ,e_j\rangle |\Vert e_j\Vert _{L^p}\le \Vert e_j\Vert _{L^p}\Vert e_j\Vert _{L^{p^{\prime }}}\Vert \phi \Vert _{L^p}, \ \ 1\le p\le \infty .\) 
by Hölder's inequality. Note that \(P_{\lambda _j} e_j=\lambda _j e_j\) , hence by scattering theory (see e.g. Hörmander [3]}), we can obtain that \(e_j\)  is a rapidly decreasing eigenfunction, i.e.
\({\langle }x{\rangle }^\ell \partial _x^k e_j\in L^2({\mathbb {R}})\ \text{for all $\ell \in {\mathbb {N}}$ and $0\le k\le 2$}.\) 

In particular, \(e_j\in L^\infty (\mathbb {R})\)  by Sobolev's embedding. Moreover,
Hölder's inequality implies
\(\Vert e_j\Vert _{L^1}\lesssim \Vert {\langle }x{\rangle }e_j\Vert _{L^2}<\infty \) . Hence \(e_j\in L^p({\mathbb {R}})\)  for all \(1\le p\le \infty \)  by interpolation.
